Spectral instability and non-uniqueness for the Keller-Segel system

We show that the Cauchy problem associated with the parabolic-elliptic Keller-Segel model is locally ill-posed in $L^q(\mathbb{R}^n)$ for dimensions $n \in \{3,\dots,9\}$ and throughout the supercritical range $q\in [1,\frac{n}{2})$.

An analog non-uniqueness result is given in the critical space of bounded functions taking values in $L^{n/2,\infty}(\R^n)$.

The non-uniqueness is driven by an instability mechanism in self-similarity variables, in the spirit of the program proposed by Jia and Šverák for the three-dimensional Navier-Stokes equations.